Properties

Label 2808.47
Modulus $2808$
Conductor $1404$
Order $36$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2808, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,0,14,9]))
 
pari: [g,chi] = znchar(Mod(47,2808))
 

Basic properties

Modulus: \(2808\)
Conductor: \(1404\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1404}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2808.gs

\(\chi_{2808}(47,\cdot)\) \(\chi_{2808}(239,\cdot)\) \(\chi_{2808}(551,\cdot)\) \(\chi_{2808}(671,\cdot)\) \(\chi_{2808}(983,\cdot)\) \(\chi_{2808}(1175,\cdot)\) \(\chi_{2808}(1487,\cdot)\) \(\chi_{2808}(1607,\cdot)\) \(\chi_{2808}(1919,\cdot)\) \(\chi_{2808}(2111,\cdot)\) \(\chi_{2808}(2423,\cdot)\) \(\chi_{2808}(2543,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.715258917292967305309352404372968157142855191763143540395686996742363511318577676288.1

Values on generators

\((703,1405,2081,1081)\) → \((-1,1,e\left(\frac{7}{18}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2808 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2808 }(47,a) \;\) at \(\;a = \) e.g. 2